Properties of root subspaces
Let be a square matrix and let
be its eigenvalue. As we know, the nonzero elements of the null space
are the corresponding eigenvectors. This definition is generalized as follows.
Definition 1. The subspaces
are called root subspaces of
corresponding to
Exercise 1. a) Root subspaces are increasing:
(1) for all
and b) there is such that all inclusions (1) are strict for
and
(2)
Proof. a) If for some
then
which shows that
b) (1) implies Since all root subspaces are contained in
there are
such that
Let
be the smallest such
Then all inclusions (1) are strict for
Suppose for some
Then there exists
such that
, that is,
Put
Then
This means
that which contradicts the definition of
Definition 2. Property (2) can be called stabilization. The number from (2) is called a height of the eigenvalue
.
Exercise 2. Let and let
be the number from Exercise 1. Then
(3)
Proof. By the rank-nullity theorem applied to we have
By Exercise 3, to prove (3) it is sufficient to establish that
Let's assume that
contains a nonzero vector
Then we have
for some
We obtain two facts:
It follows that is a nonzero element of
This contradicts (2). Hence, the assumption
is wrong, and (3) follows.
Exercise 3. Both subspaces at the right of (3) are invariant with respect to
Proof. If then by commutativity of
and
we have
so
Suppose so that
for some
Then
Exercise 3 means that, for the purpose of further analyzing we can consider its restrictions onto
and
Exercise 4. The restriction of onto
does not have eigenvalues other than
Proof. Suppose
for some
Since
we have
Then
and
. This implies
Exercise 5. The restriction of onto
does not have
as an eigenvalue (so that
is invertible).
Proof. Suppose and
Then
for some
and
By Exercise 1
and
This contradicts the choice of
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